There is an overlap in membership, but that doesn't always extend to policy: the ACLU sued the Obama administration over immigration, and they have backed up Rush Limbaugh, Oliver North, and the KKK in the past.
(There is also something on healthcare that appears similar, but I'll freely admit I don't understand enough about the background there and can't get past the paywall without giving up data: https://www.wsj.com/articles/wheres-the-aclu-when-you-need-i...)
Have it focus on the issues and not the parties.
That's what they have to do anyway if they want to be effective. At any given time more often than not -- including right now -- the same party doesn't control both houses of Congress and the whitehouse, so anything you want to do needs bipartisan support anyway.
What's really happening here is that organizations that are supposed to be about issues are getting coopted by partisans -- which makes them less effective at their stated purpose, but more useful to the partisans who can then remove support from the opposing party members who actually support that issue.
As for how "you" can do it, don't support "issue organizations" that are really partisan organizations, and instead support actual issue organizations. You can tell the difference because you can see real issue organizations regularly working with both parties and supporting candidates who support their issues regardless of which party they're in.
https://www.aclu.org/other/fix-fisa-end-warrantless-wiretapp...
They don't give a fsck about the political inclination of those speaking. They even fought for the rights of Nazis to march in Skokie:
https://www.aclu.org/other/aclu-history-taking-stand-free-sp...
https://www.wsj.com/articles/the-aclu-retreats-from-free-exp...